scale networks - rensselaer polytechnic...

60 S C I E N T I F I C A M E R I C A N M A Y 2 0 0 3

Scientists have recently discovered that various complex systems have an underlying architecture governed by shared organizing principles.

This insight has important implications for a host of applications, from drug development to Internet security

BY ALBERT-LÁSZLÓ BARABÁSI AND ERIC BONABEAU

Scale-FreeNetworks

THE INTERNET, mapped on the opposite page, is a scale-free network in thatsome sites (starbursts and detail above) have a seemingly unlimited

number of connections to other sites. This map, made on February 6, 2003,traces the shortest routes from a test Web site to about 100,000 others,

using like colors for similar Web addresses.

COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC.

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The brain is a network of nerve cells con-nected by axons, and cells themselves arenetworks of molecules connected by bio-chemical reactions. Societies, too, are net-works of people linked by friendships,familial relationships and professionalties. On a larger scale, food webs and eco-systems can be represented as networksof species. And networks pervade tech-nology: the Internet, power grids andtransportation systems are but a few ex-amples. Even the language we are usingto convey these thoughts to you is a net-work, made up of words connected bysyntactic relationships.

Yet despite the importance and per-vasiveness of networks, scientists havehad little understanding of their structureand properties. How do the interactionsof several malfunctioning nodes in a com-plex genetic network result in cancer?How does diffusion occur so rapidly incertain social and communications sys-tems, leading to epidemics of diseases andcomputer viruses? How do some net-works continue to function even after thevast majority of their nodes have failed?

Recent research has begun to answersuch questions. Over the past few years,investigators from a variety of fields havediscovered that many networks—fromthe World Wide Web to a cell’s metabol-ic system to actors in Hollywood—aredominated by a relatively small numberof nodes that are connected to many oth-er sites. Networks containing such im-portant nodes, or hubs, tend to be whatwe call “scale-free,” in the sense thatsome hubs have a seemingly unlimited

number of links and no node is typical ofthe others. These networks also behave incertain predictable ways; for example,they are remarkably resistant to acciden-tal failures but extremely vulnerable tocoordinated attacks.

Such discoveries have dramaticallychanged what we thought we knew aboutthe complex interconnected world aroundus. Unexplained by previous network the-ories, hubs offer convincing proof thatvarious complex systems have a strict ar-chitecture, ruled by fundamental laws—

laws that appear to apply equally to cells,computers, languages and society. Fur-thermore, these organizing principles havesignificant implications for developingbetter drugs, defending the Internet fromhackers, and halting the spread of deadlyepidemics, among other applications.

Networks without ScaleFOR MORE THAN 40 YEARS, sciencetreated all complex networks as beingcompletely random. This paradigm has itsroots in the work of two Hungarian math-ematicians, the inimitable Paul Erd ″os andhis close collaborator Alfréd Rényi. In1959, aiming to describe networks seen incommunications and the life sciences, Erd″os and Rényi suggested that such sys-tems could be effectively modeled by con-necting their nodes with randomly placedlinks. The simplicity of their approach andthe elegance of some of their related theo-rems revitalized graph theory, leading tothe emergence of a field in mathematicsthat focuses on random networks.

An important prediction of random-

network theory is that, despite the ran-dom placement of links, the resulting sys-tem will be deeply democratic: mostnodes will have approximately the samenumber of links. Indeed, in a random net-work the nodes follow a Poisson distrib-ution with a bell shape, and it is extreme-ly rare to find nodes that have significant-ly more or fewer links than the average.Random networks are also called expo-nential, because the probability that anode is connected to k other sites de-creases exponentially for large k.

So in 1998, when we, together withHawoong Jeong and Réka Albert of theUniversity of Notre Dame, embarked ona project to map the World Wide Web,we expected to find a random network.Here’s why: people follow their uniqueinterests when deciding what sites to linktheir Web documents to, and given the di-versity of everyone’s interests and thetremendous number of pages they canchoose from, the resulting pattern of con-nections should appear fairly random.

The measurements, however, defiedthat expectation. Software designed forthis project hopped from one Web pageto another and collected all the links itcould. Although this virtual robot reachedonly a tiny fraction of the entire Web, themap it assembled revealed somethingquite surprising: a few highly connectedpages are essentially holding the WorldWide Web together. More than 80 per-cent of the pages on the map had fewerthan four links, but a small minority, lessthan 0.01 percent of all nodes, had morethan 1,000. (A subsequent Web surveywould uncover one document that hadbeen referenced by more than two millionother pages!)

Counting how many Web pages haveexactly k links showed that the distribu-tion followed a so-called power law: theprobability that any node was connectedto k other nodes was proportional to 1⁄ kn.The value of n for incoming links was ap-proximately 2, so, for instance, any nodewas roughly four times as likely to havejust half the number of incoming links asanother node. Power laws are quite dif-ferent from the bell-shaped distributionsthat characterize random networks.


■ A variety of complex systems share an important property: some nodes have a tremendous number of connections to other nodes, whereas most nodes havejust a handful. The popular nodes, called hubs, can have hundreds, thousands or even millions of links. In this sense, the network appears to have no scale.

■ Scale-free networks have certain important characteristics. They are, for instance, robust against accidental failures but vulnerable to coordinated attacks.

■ Understanding of such characteristics could lead to new applications in many arenas. For example, computer scientists might be able to devise more effective strategies for preventing computer viruses from crippling a network such as the Internet.

Overview/Scale-Free Networks

Networks are everywhere.


Specifically, a power law does not have apeak, as a bell curve does, but is instead de-scribed by a continuously decreasing func-tion. When plotted on a double-logarith-mic scale, a power law is a straight line[see illustration above]. In contrast to thedemocratic distribution of links seen inrandom networks, power laws describesystems in which a few hubs, such as Ya-hoo and Google, dominate.

Hubs are simply forbidden in randomnetworks. When we began to map theWeb, we expected the nodes to follow abell-shaped distribution, as do people’sheights. Instead we discovered certainnodes that defied explanation, almost asif we had stumbled on a significant num-ber of people who were 100 feet tall, thusprompting us to coin the term “scale-free.”

Scale-Free Networks AboundOVER THE PAST several years, re-searchers have uncovered scale-free struc-tures in a stunning range of systems.When we studied the World Wide Web,we looked at the virtual network of Webpages connected to one another by hy-perlinks. In contrast, Michalis Faloutsosof the University of California at River-side, Petros Faloutsos of the University ofToronto and Christos Faloutsos of Car-negie Mellon University analyzed thephysical structure of the Internet. Thesethree computer-scientist brothers investi-gated the routers connected by optical orother communications lines and foundthat the topology of that network, too, isscale-free.

Researchers have also discovered that

some social networks are scale-free. A col-laboration between scientists from BostonUniversity and Stockholm University, forinstance, has shown that a network ofsexual relationships among people inSweden followed a power law: althoughmost individuals had only a few sexualpartners during their lifetime, a few (thehubs) had hundreds. A recent study ledby Stefan Bornholdt of the University ofKiel in Germany concluded that the net-work of people connected by e-mail islikewise scale-free. Sidney Redner ofBoston University demonstrated that thenetwork of scientific papers, connectedby citations, follows a power law as well.And Mark Newman of the University ofMichigan at Ann Arbor examined col-laborations among scientists in several

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RANDOM VERSUS SCALE-FREE NETWORKS

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RANDOM NETWORKS, which resemble the U.S. highway system(simplified in left map), consist of nodes with randomly placedconnections. In such systems, a plot of the distribution of nodelinkages will follow a bell-shaped curve (left graph), with mostnodes having approximately the same number of links.

In contrast, scale-free networks, which resemble the U.S.airline system (simplified in right map), contain hubs (red)—

nodes with a very high number of links. In such networks, thedistribution of node linkages follows a power law (center graph)in that most nodes have just a few connections and some have a tremendous number of links. In that sense, the system has no“scale.” The defining characteristic of such networks is that thedistribution of links, if plotted on a double-logarithmic scale(right graph), results in a straight line.

Bell Curve Distribution of Node Linkages Power Law Distribution of Node Linkages

Random Network Scale-Free Network


disciplines, including physicians and com-puter scientists, and found that those net-works were also scale-free, corroboratinga study we conducted focusing on math-ematicians and neurologists. (Interesting-ly, one of the largest hubs in the mathe-matics community is Erd″os himself, whowrote more than 1,400 papers with nofewer than 500 co-authors.)

Scale-free networks can occur in busi-ness. Walter W. Powell of Stanford Uni-versity, Douglas R. White of the Univer-sity of California at Irvine, Kenneth W.Koput of the University of Arizona, andJason-Owen Smith of the University ofMichigan studied the formation of al-liance networks in the U.S. biotechnolo-gy industry and discovered definite hubs—

for instance, companies such as Genzyme,Chiron and Genentech had a dispropor-tionately large number of partnershipswith other firms. Researchers in Italy tooka deeper look at that network. Using datacollected by the University of Siena’s Phar-maceutical Industry Database, which nowprovides information for around 20,100R&D agreements among more than 7,200organizations, they found that the hubsdetected by Powell and his colleagues wereactually part of a scale-free network.

Even the network of actors in Holly-wood—popularized by the game Six De-grees of Kevin Bacon, in which playerstry to connect actors to Bacon via themovies in which they have appeared to-gether—is scale-free. A quantitative analy-

sis of that network showed that it, too,is dominated by hubs. Specifically, al-though most actors have only a few linksto others, a handful of actors, includingRod Steiger and Donald Pleasence, havethousands of connections. (Incidentally,on a list of most connected actors, Baconranked just 876th.)

On a more serious note, scale-freenetworks are present in the biologicalrealm. With Zoltán Oltvai, a cell biologistfrom Northwestern University, we founda scale-free structure in the cellular meta-bolic networks of 43 different organismsfrom all three domains of life, includingArchaeoglobus fulgidus (an archaebac-terium), Escherichia coli (a eubacterium)and Caenorhabditis elegans (a eukary-ote). In such networks, cells burn food bysplitting complex molecules to release en-ergy. Each node is a particular molecule,and each link is a biochemical reaction.We found that most molecules participatein just one or two reactions, but a few (thehubs), such as water and adenosine tri-phosphate, play a role in most of them.

We discovered that the protein-inter-action network of cells is scale-free as well.In such a network, two proteins are “con-nected” if they are known to interact witheach other. When we investigated Baker’syeast, one of the simplest eukaryotic (nu-cleus-containing) cells, with thousands ofproteins, we discovered a scale-free topol-ogy: although most proteins interact withonly one or two others, a few are able to

attach themselves physically to a hugenumber. We found a similar result in theprotein-interaction network of an organ-ism that is very different from yeast, a sim-ple bacterium called Helicobacter pylori.

Indeed, the more that scientists stud-ied networks, the more they uncoveredscale-free structures. These findings raisedan important question: How can systemsas fundamentally different as the cell andthe Internet have the same architectureand obey the same laws? Not only arethese various networks scale-free, theyalso share an intriguing property: for rea-sons not yet known, the value of n in thekn term of the power law tends to fall be-tween 2 and 3.

The Rich Get RicherPERHAPS A MORE BASIC question iswhy random-network theory fails to ex-plain the existence of hubs. A closer ex-amination of the work of Erd″os and Rén-yi reveals two reasons.

In developing their model, Erd″os andRényi assumed that they had the full in-ventory of nodes before they placed thelinks. In contrast, the number of docu-ments on the Web is anything but con-stant. In 1990 the Web had only one page.Now it has more than three billion. Mostnetworks have expanded similarly. Hol-lywood had only a handful of actors in1890, but as new people joined the trade,the network grew to include more thanhalf a million, with the rookies connect-ing to veteran actors. The Internet hadonly a few routers about three decadesago, but it gradually grew to have mil-lions, with the new routers always linkingto those that were already part of the net-work. Thanks to the growing nature ofreal networks, older nodes had greateropportunities to acquire links.

Furthermore, all nodes are not equal.When deciding where to link their Webpage, people can choose from a few billionlocations. Yet most of us are familiar withonly a tiny fraction of the full Web, andthat subset tends to include the more con-nected sites because they are easier to find.By simply linking to those nodes, peopleexercise and reinforce a bias toward them.This process of “preferential attachment”occurs elsewhere. In Hollywood the more


NETWORK NODES LINKS

Cellular metabolism Molecules involved in Participation in the sameburning food for energy biochemical reaction

Hollywood Actors Appearance in the same movie

Internet Routers Optical and other physical connections

Protein regulatory Proteins that help to Interactions among network regulate a cell’s activities proteins

Research collaborations Scientists Co-authorship of papers

Sexual relationships People Sexual contact

World Wide Web Web pages URLs

Examples of Scale-Free Networks


connected actors are more likely to bechosen for new roles. On the Internet themore connected routers, which typicallyhave greater bandwidth, are more desir-able for new users. In the U.S. biotech in-dustry, well-established companies such asGenzyme tend to attract more alliances,which further increases their desirabilityfor future partnerships. Likewise, themost cited articles in the scientific litera-ture stimulate even more researchers toread and cite them, a phenomenon thatnoted sociologist Robert K. Mertoncalled the Matthew effect, after a passagein the New Testament: “For unto everyone that hath shall be given, and he shallhave abundance.”

These two mechanisms—growth andpreferential attachment—help to explainthe existence of hubs: as new nodes ap-pear, they tend to connect to the moreconnected sites, and these popular loca-tions thus acquire more links over timethan their less connected neighbors. Andthis “rich get richer” process will gener-ally favor the early nodes, which are morelikely to eventually become hubs.

Along with Réka Albert, we have usedcomputer simulations and calculations toshow that a growing network with pref-erential attachment will indeed becomescale-free, with its distribution of nodesfollowing a power law. Although this the-oretical model is simplistic and needs tobe adapted to specific situations, it doesappear to confirm our explanation for

why scale-free networks are so ubiquitousin the real world.

Growth and preferential attachmentcan even help explicate the presence ofscale-free networks in biological systems.Andreas Wagner of the University ofNew Mexico and David A. Fell of OxfordBrookes University in England havefound, for instance, that the most-con-nected molecules in the E. coli metabolicnetwork tend to have an early evolution-ary history: some are believed to be rem-nants of the so-called RNA world (theevolutionary step before the emergence ofDNA), and others are components of themost ancient metabolic pathways.

Interestingly, the mechanism of pref-erential attachment tends to be linear. Inother words, a new node is twice as like-ly to link to an existing node that hastwice as many connections as its neigh-bor. Redner and his colleagues at BostonUniversity and elsewhere have investigat-ed different types of preferential attach-ment and have learned that if the mecha-nism is faster than linear (for example, anew node is four times as likely to link to

an existing node that has twice as manyconnections), one hub will tend to runaway with the lion’s share of connections.In such “winner take all” scenarios, thenetwork eventually assumes a star topol-ogy with a central hub.

An Achilles’ HeelAS HUMANITY BECOMES increasing-ly dependent on power grids and com-munications webs, a much-voiced con-cern arises: Exactly how reliable are thesetypes of networks? The good news is thatcomplex systems can be amazingly re-silient against accidental failures. In fact,although hundreds of routers routinelymalfunction on the Internet at any mo-ment, the network rarely suffers majordisruptions. A similar degree of robust-ness characterizes living systems: peoplerarely notice the consequences of thou-sands of errors in their cells, ranging frommutations to misfolded proteins. What isthe origin of this robustness?

Intuition tells us that the breakdownof a substantial number of nodes will re-sult in a network’s inevitable fragmenta-


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BIRTH OF A SCALE-FREE NETWORKA SCALE-FREE NETWORK grows incrementally from two to 11 nodes in this example. When deciding where to establish a link, a new node(green) prefers to attach to an existing node (red) that already has many other connections. These two basic mechanisms—growthand preferential attachment—will eventually lead to the system’s being dominated by hubs, nodes having an enormous number of links.

ALBERT-LÁSZLÓ BARABÁSI and ERIC BONABEAU study the behavior and characteristics ofmyriad complex systems, ranging from the Internet to insect colonies. Barabási is Emil T.Hofman Professor of Physics at the University of Notre Dame, where he directs researchon complex networks. He is author of Linked: The New Science of Networks. Bonabeau ischief scientist at Icosystem, a consulting firm based in Cambridge, Mass., that applies thetools of complexity science to the discovery of business opportunities. He is co-author ofSwarm Intelligence: From Natural to Artificial Systems. This is Bonabeau’s second articlefor Scientific American.

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tion. This is certainly true for random net-works: if a critical fraction of nodes is re-moved, these systems break into tiny,noncommunicating islands. Yet simula-tions of scale-free networks tell a differentstory: as many as 80 percent of random-ly selected Internet routers can fail and theremaining ones will still form a compactcluster in which there will still be a pathbetween any two nodes. It is equally dif-ficult to disrupt a cell’s protein-interactionnetwork: our measurements indicate thateven after a high level of random muta-tions are introduced, the unaffected pro-teins will continue to work together.

In general, scale-free networks dis-play an amazing robustness against ac-cidental failures, a property that is root-ed in their inhomogeneous topology. Therandom removal of nodes will take outmainly the small ones because they aremuch more plentiful than hubs. And theelimination of small nodes will not dis-rupt the network topology significantly,because they contain few links comparedwith the hubs, which connect to nearlyeverything. But a reliance on hubs has a serious drawback: vulnerability to attacks.

In a series of simulations, we found

that the removal of just a few key hubsfrom the Internet splintered the systeminto tiny groups of hopelessly isolatedrouters. Similarly, knockout experimentsin yeast have shown that the removal ofthe more highly connected proteins has asignificantly greater chance of killing theorganism than does the deletion of othernodes. These hubs are crucial—if muta-tions make them dysfunctional, the cellwill most likely die.

A reliance on hubs can be advanta-geous or not, depending on the system.Certainly, resistance to random break-down is good news for both the Internetand the cell. In addition, the cell’s relianceon hubs provides pharmaceutical re-searchers with new strategies for selectingdrug targets, potentially leading to curesthat would kill only harmful cells or bac-teria by selectively targeting their hubs,while leaving healthy tissue unaffected.But the ability of a small group of well-in-formed hackers to crash the entire com-munications infrastructure by targetingits hubs is a major reason for concern.

The Achilles’ heel of scale-free net-works raises a compelling question: Howmany hubs are essential? Recent research

suggests that, generally speaking, the si-multaneous elimination of as few as 5 to15 percent of all hubs can crash a system.For the Internet, our experiments implythat a highly coordinated attack—first re-moving the largest hub, then the nextlargest, and so on—could cause signifi-cant disruptions after the elimination ofjust several hubs. Therefore, protectingthe hubs is perhaps the most effectiveway to avoid large-scale disruptionscaused by malicious cyber-attacks. Butmuch more work is required to deter-mine just how fragile specific networksare. For instance, could the failure of sev-eral hubs like Genzyme and Genentechlead to the collapse of the entire U.S. bio-tech industry?

Scale-Free EpidemicsK N O W L E D G E A B O U T scale-free net-works has implications for understandingthe spread of computer viruses, diseasesand fads. Diffusion theories, intensivelystudied for decades by both epidemi-ologists and marketing experts, predict acritical threshold for the propagation ofa contagion throughout a population.Any virus, disease or fad that is less in-fectious than that well-defined thresholdwill inevitably die out, whereas thoseabove the threshold will multiply expo-nentially, eventually penetrating the en-tire system.

Recently, though, Romualdo Pastor-Satorras of the Polytechnic University ofCatalonia in Barcelona and AlessandroVespigniani of the International Centerfor Theoretical Physics in Trieste, Italy,reached a disturbing conclusion. Theyfound that in a scale-free network thethreshold is zero. That is, all viruses, eventhose that are weakly contagious, willspread and persist in the system. This re-sult explains why Love Bug, the mostdamaging computer virus thus far (it shutdown the British Parliament in 2000),was still one of the most pervasive virus-es a year after its supposed eradication.

Because hubs are connected to manyother nodes, at least one hub will tend tobe infected by any corrupted node. Andonce a hub has been infected, it will passthe virus to numerous other sites, eventu-ally compromising other hubs, which will


Computing■ Computer networks with scale-free architectures, such as the World Wide Web,

are highly resistant to accidental failures. But they are very vulnerable to deliberateattacks and sabotage.

■ Eradicating viruses, even known ones, from the Internet will be effectively impossible.

Medicine■ Vaccination campaigns against serious viruses, such as smallpox, might be most

effective if they concentrate on treating hubs—people who have many connections to others. But identifying such individuals can be difficult.

■ Mapping out the networks within the human cell could aid researchers in uncoveringand controlling the side effects of drugs. Furthermore, identifying the hub moleculesinvolved in certain diseases could lead to new drugs that would target those hubs.

Business■ Understanding how companies, industries and economies are interlinked could help

researchers monitor and avoid cascading financial failures.■ Studying the spread of a contagion on a scale-free network could offer new ways for

marketers to propagate consumer buzz about their products.

The Potential Implicationsof Scale-Free Networks for . . .


then spread the virus throughout the en-tire system.

The fact that biological viruses spreadin social networks, which in many casesappear to be scale-free, suggests that sci-entists should take a second look at thevolumes of research written on the inter-play of network topology and epidemics.Specifically, in a scale-free network, the

traditional public health approach of ran-dom immunization could easily fail be-cause it would very likely neglect a num-ber of the hubs. In fact, nearly everyonewould have to be treated to ensure thatthe hubs were not missed. A vaccinationfor measles, for instance, must reach 90percent of the population to be effective.

Instead of random immunizations,

though, what if doctors targeted the hubs,or the most connected individuals? Re-search in scale-free networks indicatesthat this alternative approach could be ef-fective even if the immunizations reachedonly a small fraction of the overall popu-lation, provided that the fraction con-tained the hubs.

But identifying the hubs in a social

HOW ROBUST ARE RANDOM AND SCALE-FREE NETWORKS?

THE ACCIDENTAL FAILURE of a number of nodes in a randomnetwork (top panels) can fracture the system into non-communicating islands. In contrast, scale-free networks are

more robust in the face of such failures (middle panels). But they are highly vulnerable to a coordinated attack againsttheir hubs (bottom panels).


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Scale-Free Network, Accidental Node Failure

Scale-Free Network, Attack on Hubs

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network is much more difficult than de-tecting them in other types of systems.Nevertheless, Reuven Cohen and ShlomoHavlin of Bar-Ilan University in Israel, to-gether with Daniel ben-Avraham ofClarkson University, have proposed aclever solution: immunize a small fractionof the random acquaintances of arbitrar-ily selected individuals, a procedure thatselects hubs with a high probability be-cause they are linked to many people.That approach, though, leads to a num-ber of ethical dilemmas. For instance,even if the hubs could be identified,should they have priority for immuniza-

tions and cures? Such issues notwith-standing, targeting hubs could be themost pragmatic solution for the futuredistribution of AIDS or smallpox vaccinesin countries and regions that do not havethe resources to treat everyone.

In many business contexts, peoplewant to start, not stop, epidemics. Viralmarketing campaigns, for instance, oftenspecifically try to target hubs to speed theadoption of a product. Obviously, such astrategy is not new. Back in the 1950s, astudy funded by pharmaceutical giantPfizer discovered the important role thathubs play in how quickly a community of

doctors begins using a new drug. Indeed,marketers have intuitively known forsome time that certain customers outshineothers in spreading promotional buzzabout products and fads. But recent workin scale-free networks provides the scien-tific framework and mathematical tools toprobe that phenomenon more rigorously.

From Theory to PracticeALTHOUGH SCALE-FREE networksare pervasive, numerous prominent ex-ceptions exist. For example, the highwaysystem and power grid in the U.S. are notscale-free. Neither are most networks


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IN 1967 STANLEY MILGRAM, a social psychologist at HarvardUniversity, sent hundreds of letters to people in Nebraska,asking them to forward the correspondence to acquaintanceswho might be able to shepherd it closer to a target recipient: astockbroker in Boston. To track each of the different paths,Milgram asked the participants to mail a postcard back to himwhen they passed the letter to someone else. Milgram found thatthe letters that eventually arrived at the final destination hadpassed through an average of six individuals—the basis of thepopular notion of “six degrees of separation” between everyone.

Although Milgram’s work was hardly conclusive—most of theletters never made their way to the stockbroker—scientists haverecently learned that other networks exhibit this “small world”property. We have found, for instance, that a path of just threereactions will connect almost any pair of chemicals in a cell. And

on the World Wide Web, which contains more than three billiondocuments, Web pages are typically 19 clicks from one another.

The small-world property does not necessarily indicate thepresence of any magic organizing principle. Even a large networkwith purely random connections will be a small world. Considerthat you might have about 1,000 acquaintances. If each of thoseindividuals also knows another 1,000, then a million people willbe just two handshakes away from you, a billion will be just threeaway, and the earth’s entire population will be well within four.Given that fact, the notion that any two strangers in the world areseparated by an average of six degrees seems almost trivial. Butfurther investigation reveals some deeper insights.

Our simple calculation assumes that the people you know areall strangers to one another. In reality, there is much overlap.Indeed, society is fragmented into clusters of individuals havingsimilar characteristics (such as income or interests), a featurethat has been widely discussed in the sociology literaturefollowing the seminal work in the 1970s of Mark Granovetter,then a graduate student at Harvard. Clustering is also a generalproperty of many other types of networks. In 1998 Duncan Wattsand Steven Strogatz, then both at Cornell University, foundsignificant clustering in a variety of systems, from the U.S. powergrid to the neural network of the Caenorhabditis elegans worm.

At first glance, isolated clusters of highly interconnectednodes appear to run counter to the topology of scale-freenetworks, in which a number of hubs radiate throughout thesystem, linking everything. Recently, however, we have shownthat the two properties are compatible: a network can be bothhighly clustered and scale-free when small, tightly interlinkedclusters of nodes are connected into larger, less cohesive groups(left). This type of hierarchy appears to exist in a number ofsystems, from the World Wide Web (in which clusters aregroupings of Web pages devoted to the same topic) to a cell (in which clusters are teams of molecules responsible for a specific function). —A.-L.B. and E.B.

It’s a Small World, After All

HIERARCHICAL CLUSTERS, shown schematically, could include, say, Webpages on the Frank Lloyd Wright home Fallingwater (yellow), which couldbe linked to other clusters (green) focusing on Wright, famous homes orPennsylvania’s attractions. Those sites, in turn, could be connected toclusters (red) on famous architects or architecture in general.

Clusters in differenthierarchical levels


seen in materials science. In a crystal lat-tice, for instance, atoms have the samenumber of links to their neighbors. Withother networks, the data are inconclusive.The relatively small size of food webs,which show predator-prey relationships,has prevented scientists from reaching aclear conclusion regarding that network’stype. And the absence of large-scale con-nectivity maps of the brain has kept re-searchers from knowing the nature ofthat important network as well.

Determining whether a network isscale-free is important in understandingthe system’s behavior, but other signifi-cant parameters merit attention, too.One such characteristic is the diameter,or path length, of a network: the largestnumber of hops required to get from onenode to another by following the shortestroute possible [see box on opposite page].

Finally, knowledge of a network’s gen-eral topology is just part of the story in un-derstanding the overall characteristics and

behavior of such systems. There might besteep costs, for instance, with the additionof each link to a given node that couldprevent certain networks (such as the U.S.highway system) from becoming scale-free. In food chains, some prey are easierto catch than others, and that fact has aprofound effect on the overall ecosystem.With social networks, ties among house-hold members are much stronger thanconnections to casual acquaintances, sodiseases (and information) are more like-ly to spread through such linkages. Fortransportation, transmission and commu-nications systems (such as the Internet),

congestion along specific links is a majorconsideration: too much traffic on a par-ticular link can cause it to break down,leading to the potential failure of otherlinks that must then handle the spillover.And the nodes themselves might not behomogeneous—certain Web pages havemore interesting content, for instance—

which could greatly alter the preferential-attachment mechanism.

Because of these and other factors, sci-entists have only begun to uncover the be-havior of scale-free systems. Immunizinghubs, for instance, might not be sufficientto stop the spread of a disease; a more ef-fective solution might be found by con-sidering not just the number of connec-tions a person has but also the frequencyand duration of contact for those links.

In essence, we have studied complexnetworks first by ignoring the details oftheir individual links and nodes. By dis-tancing ourselves from those particulars,we have been able to better glimpse someof the organizing principles behind theseseemingly incomprehensible systems. Atthe very least, knowledge from this en-deavor has led to the rethinking of manybasic assumptions. In the past, for exam-ple, researchers modeled the Internet as arandom network to test how a new rout-ing protocol might affect system conges-tion. But we now know that the Internet isa scale-free system with behavior that isdramatically different from a random net-work’s. Consequently, investigators suchas John W. Byers and his colleagues atBoston University are revamping the com-puter models they have been using to sim-ulate the Internet. Similarly, knowledge ofthe properties of scale-free networks willbe valuable in a number of other fields, es-pecially as we move beyond network to-pologies to probe the intricate and oftensubtle dynamics taking place within thosecomplex systems.


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All the World’s a Net. David Cohen in New Scientist, Vol. 174, No. 2338, pages 24–29; April 13, 2002.

Statistical Mechanics of Complex Networks. Réka Albert and Albert-László Barabási in Reviews of Modern Physics, Vol. 74, pages 47–97; January 2002.

Linked: The New Science of Networks. Albert-László Barabási. Perseus Publishing, 2002.

Evolution of Networks: From Biological Nets to the Internet and WWW. J.F.F. Mendes and Sergei N.Dorogovtsev. Oxford University Press, 2003.

Find links to papers on scale-free networks at www.nd.edu/~networks

M O R E T O E X P L O R E

MAP OF INTERACTING PROTEINS in yeast highlights the discovery that highly linked, or hub, proteinstend to be crucial for a cell’s survival. Red denotes essential proteins (their removal will cause the cellto die). Orange represents proteins of some importance (their removal will slow cell growth). Greenand yellow represent proteins of lesser or unknown significance, respectively.


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